{"id":"c56ecf0d-b66d-4ae6-a0d2-5f17669e603a","arxiv_id":"2501.15033","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For indefinite anisotropic quadratic forms in three variables with td(f) square-free, once a single integer solution exists, there are infinitely many solutions with one coordinate having at most 6 prime factors (5 under Selberg's conjecture).","lead":"This math paper proves that certain three-variable quadratic equations have infinitely many whole-number solutions where one coordinate has at most six prime factors. It improves previous results in the study of equations that are expected to have prime-number solutions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2 relies on an unstated two-variable analogue of Lemma 2.3; Lemma 2.3 only treats x1, while sieving x1x2 modulo d requires joint (x1,x2) equidistribution.","rationale":"The reader correctly identifies Lemma 2.3 as the key sieve input and worries about its derivation from [18]. I agree that the one-coordinate P6 result depends on it. However, the more concrete gap in the written proof is that Theorem 1.2, a central claim of the paper, needs a two-variable equidistribution estimate for the sequence x1x2, and no such lemma appears in Section 2. The statement in Section 2 that \"the situation for two variables is similar\" may well be true, and the same spectral method probably supplies it, but it is not written down. If the required joint estimate is provided with the same error dependence, the proof of Theorem 1.2 goes through and the paper should be accepted. If the joint estimate has a worse dependence on d, the claimed constants 16 and 14 are not reached. Thus the appropriate verdict is conditional: accept once the two-variable analogue of Lemma 2.3 is stated and proved, or shown to follow from the methods of [18] with the stated error term.","tokens_in":11204,"tokens_out":24159,"duration_ms":218106,"concrete_test":"State and prove the two-variable analogue of Lemma 2.3: for a_n^{(2)}(T) = Σ_{x∈O, x1x2=±n} F_T(x), derive Σ_{n≡0(d)} a_n^{(2)}(T) = (ω2(d)/d) X + O(d^{A+ε} T^{1/2+θ+ε}) with ω2(p)/p = 2/p + O(1/p^2). Determine the minimal exponent A; the single-coordinate proof suggests A=1, but the natural count of the joint congruence set may give A=2. Recompute the level-of-distribution bound Σ_{d<D} 4^{ν(d)} |R_d| ≤ X/log^3 X with this A; if A=2, the attainable τ drops below 25/128 and the minimum m(ζ) in Section 4.2 exceeds 16, so Theorem 1.2 would need a different argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 2 proves Lemma 2.3 only for the sequence a_n(T) with x1 = ±n, i.e. a one-coordinate congruence. Section 4.2 then runs the dimension-2 weighted sieve on the product x1x2 and uses \"τ = 25/128 as before\". This is not licensed by Lemma 2.3: the condition d | x1x2 expands, for squarefree d, into a sum over assignments of each prime p|d to p|x1, p|x2, or both. Controlling that sum requires a joint equidistribution statement for the pair (x1,x2) modulo d, with local density ω2(p)/p = 2/p + O(1/p^2) and an error term compatible with (3.2). No such lemma is stated or proved. If the joint estimate only has error O(d^{2+ε} T^{1/2+θ+ε}) rather than O(d^{1+ε} T^{1/2+θ+ε}), the divisor sum in (3.2) forces a smaller level of distribution than 25/128, and the computation m(0.19214...) ≈ 15.6327 in Section 4.2 no longer yields r = 16. This is a missing proof obligation, not a demonstrated contradiction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two almost-prime results for integral points on a three-variable indefinite anisotropic quadratic form f(x)=t with td(f) square-free. Assuming V(Z) is non-empty, Theorem 1.1 states that the set of points with x1 having at most 6 prime factors outside the fixed bad set B={2,3,5,7} is Zariski dense, with 6 improved to 5 under Selberg's eigenvalue conjecture. Theorem 1.2 states the analogous Zariski-density result for the product x1x2 having at most 16 prime factors outside B, with 16 improved to 14 under Selberg's conjecture. The proofs combine an equidistribution estimate for the orbit (Lemma 2.3), Kim-Sarnak bounds via Jacquet-Langlands, and explicit one- and two-dimensional weighted sieves of Diamond-Halberstam and Halberstam-Richert.","tokens_in":55,"tokens_out":5589,"duration_ms":113952,"significance":"If fully correct, the results are a genuine improvement over the earlier Liu-Sarnak theorem, replacing almost-primality of the product of all three coordinates with almost-primality of one or two coordinates. The numerical sieve computations are explicit, and the claimed constants 6 and 16 are not borrowed from prior work but arise from the stated level of distribution. The main deficiency is that Theorem 1.2 relies on an unstated and unproved two-variable equidistribution estimate, so the significance of the paper's second theorem depends on an external claim that the authors should be asked to justify.","major_comments":[{"comment":"Theorem 1.2 sieves the product x1x2, but Lemma 2.3 is proved only for the sequence a_n(T) defined in (2.4) by x1=±n, so the congruence n≡0 mod d in (2.8) corresponds to d | x1. The two-dimensional sieve in §4.2 requires an analogue for a sequence such as b_n(T)=sum_{x1x2=±n} F_T(x), with a distribution estimate for d | x1x2. The sentence in §2 that 'the situation for two variables is similar' is not a proof, and no such lemma is stated or proved. This is load-bearing: the constants 16 and 14 in §4.2 are computed from (3.2) with τ=25/128 or τ=1/4, and those computations are only valid for a sequence whose equidistribution error has the same shape as (2.8). A joint estimate with error O(d^{2+ε} T^{1/2+θ+ε}) would change the admissible level of distribution and the numerical values of m(ζ). Please supply a complete statement and proof of the two-variable equidistribution, or an explicit derivation from [18].","section":"§4.2 and §2.2"},{"comment":"The application of the nonlinear sieve in Lemma 3.2 is not fully specified: the paper does not define the finite sequence A used for Theorem 1.2, nor does it verify the two hypotheses max_{a_n∈A} n ≤ X^{τμ} and (3.2) for that sequence. For instance, the value μ=2/τ is asserted without stating what n represents for the product x1x2. This gap is partly formal, but it is connected to the missing joint equidistribution: one needs to know that the same level τ controls the error term for the product sequence before applying the sieve conclusion r>m(ζ).","section":"§4.2"}],"minor_comments":[{"comment":"There are several conversion artifacts, such as 'V ariables' in the title and 's /greaterorequalslant3' in §1.1; these should be corrected during production.","section":"Title and Abstract"},{"comment":"The notation O^0(Z/dZ) is used both for the set with y1≡0 mod d and for the set with x1x2x3≡0 mod d; the two definitions should be distinguished to avoid confusion.","section":"§2.2"},{"comment":"The displayed formula for m(ζ) contains an ambiguous term 'ζ 2 − µ β2'; adding parentheses, e.g. (2+ζ) log(β2/ζ) - 2 + (ζ^2 - μ)/β2, would clarify the algebra.","section":"§4.2"},{"comment":"The phrase 'the above 6 can be reduced to 5' and the similar phrase for 16 and 14 are grammatically awkward; 'the constant 6 can be replaced by 5' would be clearer.","section":"§1.2"}],"recommendation":"major_revision","confidential_remarks":"The central question for the editor is whether the two-variable equidistribution can be supplied. The paper's Theorem 1.1 appears sound and is a nice improvement. Theorem 1.2 is not yet fully justified in the submitted text. If the authors can state and prove the missing two-variable analogue of Lemma 2.3, the paper would be acceptable; otherwise the second theorem should be withdrawn or reworded as conditional on such an estimate. There is no indication of circularity or fitting of constants, and the numerical sieve computations themselves check out for the one-variable case."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's my read of Li–Liu, arXiv:2501.15033. The real new results are Theorem 1.1, getting P6 for a single coordinate on anisotropic ternary quadratics (P5 under Selberg), and Theorem 1.2 claiming P16 for a pair product (P14 under Selberg). Next to the prior P26 for x1x2x3 in Liu–Sarnak and P521 per coordinate in Blomer–Brüdtern, that is a genuine step. The 1-dimensional sieve for one coordinate is a genuinely new configuration, and the explicit sieve computations in §4 are careful and check out. This is a continuation of [18], but the new constants are outputs of the calculations, not inputs; no circularity.\n\nThe soft spot is Theorem 1.2. The paper never defines the sequence being sieved (the product x1x2), and never proves the required equidistribution modulo d for that product. Lemma 2.3 covers only the one-coordinate sequence with x1=±n. The condition d | x1x2 is a union of local conditions p|x1 or p|x2, and controlling the divisor sum needs a joint equidistribution statement for the pair (x1,x2). The sentence “the situation for two variables is similar” is not a proof. If the joint error term is worse than d^{1+ε}T^{1/2+θ+ε}, the level τ=25/128 used in §4.2 is unjustified, and the m(0.19214...)≈15.63 computation would not yield r=16. This is a missing proof obligation, not a demonstrated contradiction, and it is probably repairable with the same spectral machinery. But as written, Theorem 1.2 does not have a complete proof.\n\nSmaller point: Lemma 2.3 is itself stated as “the proof is exactly the same as [18, Theorem 2.1]” after a short local-density argument. That is acceptable given the identical framework, but a referee may want more detail. The Zariski-density claim is inherited from the orbit structure in the standard way, though not spelled out.\n\nWho gets value: analytic number theorists in the affine sieve program, and anyone tracking the constants in almost-prime results. It deserves a serious referee. I would send it back for revision to supply the joint equidistribution lemma and lay out the two-variable setup; if that lemma works out, this is close to a clean accept. If the joint error term turns out worse, the constants 16/14 may need adjustment.\n\nRecommendation: send to peer review, require the missing two-variable argument before acceptance.","headline":"Solid improvement on almost-prime bounds for anisotropic ternary quadratics, but Theorem 1.2 ships with an unproved two-variable equidistribution input.","tokens_in":12041,"tokens_out":5413,"would_cite":true,"duration_ms":46581,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11E20","11N36","11F72"],"pacs":[],"model":"deepseek-v4-flash","headline":"An indefinite anisotropic ternary quadratic form with one integer solution has infinitely many almost-prime solutions, with six prime factors conditionally improving to five.","keywords":["quadratic form","almost-prime","weighted sieve","Selberg eigenvalue conjecture","anisotropic quadric","spin group","Zariski density"],"falsifier":"Compute, for a concrete anisotropic form such as the norm form of a quaternion division algebra over $\\mathbb{Q}$, the weighted counts $a_n(T)$ on a fixed orbit, and compare the left side of (2.8) with $\\omega(d)X/d$ for many $d$ near $T^{25/128}$; a discrepancy exceeding the claimed $O_{\\varepsilon}(d^{1+\\varepsilon}T^{1/2+\\theta+\\varepsilon})$ would disprove the level of distribution on which the theorems rest.","tokens_in":10951,"feed_emoji":"🔢","tokens_out":14597,"duration_ms":107553,"temperature":0.7,"pith_summary":"This paper proves that an indefinite anisotropic integral quadratic form in three variables, once it has one integer solution, has infinitely many integer solutions whose coordinates are almost-primes. There are infinitely many solutions with the first coordinate having at most 6 prime factors outside the fixed set {2,3,5,7}, and infinitely many with the product of the first two coordinates having at most 16 such prime factors. If Selberg's eigenvalue conjecture holds, those counts drop to 5 and 14. The point is an almost-prime substitute for the prime-variable problem on ternary quadrics, which is currently out of reach.","feed_headline":"Six prime factors suffice on anisotropic quadrics","feed_subtitle":"One integral solution forces infinitely many with at most six prime factors; Selberg would lower it to five.","key_machinery":"The engine is an equidistribution estimate (Lemma 2.3) for the weighted number $a_n(T)$ of orbit points with first coordinate $\\pm n$, summed over $n\\equiv 0 \\pmod d$: the sum equals $\\omega(d)X/d + O_{\\varepsilon}(d^{1+\\varepsilon}T^{1/2+\\theta+\\varepsilon})$, with local density $\\omega(p)/p=1/p+O(1/p^2)$. This gives the level of distribution $\\tau = 1/4-\\theta/2 = 25/128$ using the best unconditional bound $\\theta=7/64$ toward Selberg's eigenvalue conjecture, and $\\tau=1/4$ if the conjecture holds. That level feeds a one-dimensional linear sieve for Theorem 1.1 and a two-dimensional nonlinear sieve for Theorem 1.2, whose numerical optimization produces the constants 6 and 16.","core_discovery":"Let $f(x_1,x_2,x_3)$ be an integral indefinite anisotropic quadratic form whose determinant $d(f)$ satisfies $td(f)$ square-free, and assume the congruence equation $f(x)\\equiv t \\pmod d$ is solvable for every $d$. The paper shows that the set of integer solutions with $x_1\\in P_6(B)$ is Zariski dense in the affine quadric $V$, and likewise that $\\{x: x_1x_2\\in P_{16}(B)\\}$ is Zariski dense; under Selberg's eigenvalue conjecture the constants improve to 5 and 14. Here $P_r(B)$ denotes integers with at most $r$ prime factors outside $B=\\{2,3,5,7\\}$. The proof combines the orbit structure under the spin double cover of the special orthogonal group preserving $f$, an equidistribution estimate for weighted counts of orbit points, and one- and two-dimensional weighted sieves, with spectral information coming from the quaternion algebra behind the form.","pith_inferences":["The paper does not compute the exact threshold in $\\theta$ at which $r=5$ becomes unconditional; plugging improved spectral bounds into the sieve inequalities would yield such a threshold, showing how far current technology is from the unconditional five-prime-factor result.","Because the equidistribution input is stated for an arbitrary orbit of the spin group, the same sieve framework should apply to any ternary quadric whose integral points form finitely many orbits and whose attendant quaternion algebra supplies the spectral gap; other anisotropic forms are natural test cases.","The square-free condition on $td(f)$ keeps local densities close to $1/p$; locating where the proof uses this condition would indicate whether relaxing it merely worsens the constants or changes the qualitative conclusion.","A three- or higher-dimensional analogue for products of more than two coordinates would need a higher-dimensional weighted sieve, and the nonlinear sieve machinery in the paper is already set up to handle such extensions."],"forward_implications":["For every indefinite anisotropic ternary form satisfying the hypotheses, one integral solution implies infinitely many with $x_1\\in P_6(B)$, and the set of such solutions is Zariski dense rather than confined to a subvariety.","Infinitely many solutions have $x_1x_2\\in P_{16}(B)$, so the almost-prime condition can be moved from all three coordinates to just one or two.","A proof of Selberg's eigenvalue conjecture would immediately lower the constants to 5 and 14.","Improving the unconditional spectral bound $\\theta=7/64$ would raise the level $\\tau$ and, through the same sieve inequalities, lower the constants before a full Selberg proof.","These results strengthen the earlier three-coordinate theorem $x_1x_2x_3\\in P_{26}(B)$ (22 conditional) by showing that two or even one coordinate can be left unrestricted."],"supporting_citations":[{"why":"It supplies the orbit-counting framework, the bad-prime set B, the equidistribution theorem that Lemma 2.3 adapts, and the earlier x1x2x3 almost-prime result.","marker":"[18]"},{"why":"It provides the unconditional bound theta=7/64 toward Selberg's eigenvalue conjecture that fixes the level tau=25/128.","marker":"[20]"},{"why":"It transfers the spectral gap of the quaternion algebra to the co-compact lattice Gamma via the Jacquet-Langlands correspondence.","marker":"[19]"},{"why":"It states Selberg's eigenvalue conjecture; under theta=0 the level becomes tau=1/4 and the constants drop.","marker":"[21]"},{"why":"It supplies the weighted-sieve lemmas used for the one- and two-dimensional sieves.","marker":"[6]"},{"why":"It gives the higher-dimensional sieve variant, including the kappa=1 remark used in Proposition 3.5.","marker":"[7]"},{"why":"It provides the nonlinear sieve inequality used to optimize the two-dimensional problem in Theorem 1.2.","marker":"[12]"},{"why":"It gives, through Siegel's mass formula, the local-global principle that reduces the hypothesis of one integral solution to a congruential condition.","marker":"[22]"},{"why":"It supplies the spin double cover description and the local density computation behind the bad-prime set B.","marker":"[5]"}],"fun_headline_variants":["One solution on anisotropic quadric guarantees infinitely many with ≤6 primes","Anisotropic quadrics: one solution implies infinitely many with ≤6 prime factors","Quadrics: a single integral point yields infinitely many with ≤6 primes","One solution forces infinite with ≤6 primes on anisotropic quadrics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the equidistribution estimate of Lemma 2.3: for all d up to T, the weighted solutions split across residue classes 0 modulo d with an error term of size $d^{1+\\varepsilon} T^{1/2+\\theta+\\varepsilon}$; if that error were worse, the sieve would not deliver six or sixteen prime factors.","fun_headline_variants_meta":{"raw":{"variants":["One solution on anisotropic quadric guarantees infinitely many with ≤6 primes","Anisotropic quadrics: one solution implies infinitely many with ≤6 prime factors","Quadrics: a single integral point yields infinitely many with ≤6 primes","One solution forces infinite with ≤6 primes on anisotropic quadrics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001177,"raw_usage":{"total_tokens":4854,"prompt_tokens":921,"completion_tokens":3933,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":3864}},"tokens_in":537,"tokens_out":3933,"duration_ms":40997,"temperature":1.0,"reasoning_tokens":3864,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:42:22.775130+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a concrete anisotropic form such as the norm form of a quaternion division algebra over $\\mathbb{Q}$, the weighted counts $a_n(T)$ on a fixed orbit, and compare the left side of (2.8) with $\\omega(d)X/d$ for many $d$ near $T^{25/128}$; a discrepancy exceeding the claimed $O_{\\varepsilon}(d^{1+\\varepsilon}T^{1/2+\\theta+\\varepsilon})$ would disprove the level of distribution on which the theorems rest.","supporting_citations":[{"cited_title":"Liu and P","cited_arxiv_id":null,"evidence_quote":"It supplies the orbit-counting framework, the bad-prime set B, the equidistribution theorem that Lemma 2.3 adapts, and the earlier x1x2x3 almost-prime result."},{"cited_title":"Kim and P","cited_arxiv_id":null,"evidence_quote":"It provides the unconditional bound theta=7/64 toward Selberg's eigenvalue conjecture that fixes the level tau=25/128."},{"cited_title":"Jacquet and R","cited_arxiv_id":null,"evidence_quote":"It transfers the spectral gap of the quaternion algebra to the co-compact lattice Gamma via the Jacquet-Langlands correspondence."},{"cited_title":"Selberg, On the estimation of Fourier coefficients of modular f orms, Proc","cited_arxiv_id":null,"evidence_quote":"It states Selberg's eigenvalue conjecture; under theta=0 the level becomes tau=1/4 and the constants drop."},{"cited_title":"Diamond, H","cited_arxiv_id":null,"evidence_quote":"It supplies the weighted-sieve lemmas used for the one- and two-dimensional sieves."},{"cited_title":"Proce dures for computing sieve functions","cited_arxiv_id":null,"evidence_quote":"It gives the higher-dimensional sieve variant, including the kappa=1 remark used in Proposition 3.5."},{"cited_title":"Halberstam, H","cited_arxiv_id":null,"evidence_quote":"It provides the nonlinear sieve inequality used to optimize the two-dimensional problem in Theorem 1.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives, through Siegel's mass formula, the local-global principle that reduces the hypothesis of one integral solution to a congruential condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the spin double cover description and the local density computation behind the bad-prime set B."}],"review_version":1}